Although several approaches have been adopted to study the prisoner's dilemma, including, e.g., the design of a strategy that is obtained by saturating a polynomial function in an \(n\) -population game setting [1]}, incorporating the evolution of the environment in the strategy profile through the payoff matrix poses a risk in limiting the likelihood of cooperation, and gives rise to an “oscillating tragedy of the commons” [2]}. An additional difficulty may come when we take into consideration the corresponding optional game, i.e. the optional prisoner's dilemma (OPD), where players can choose to cooperate and defect, but also abstain from playing the game [3]}. This formulation includes the dynamics of many real-life decision-making settings, such as opinion dynamics in the context of an election campaign and duopolistic competition in marketing [3]}, [5]}.
